Chemical engineering volume 1 fluid flow, heat transfer and mass transfer elsevier (1999)

Volume 1 continues to concentrate on the basic processes of Momentum Transfer as in fluid flow, Heat Transfer, and Mass Transfer, and it is also includes examples of practical applicat

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Coulson & Richardson's

Department of Chemical and Biological Process Engineering

University of Wales, Swansea

WITH

Department of Chemical and Process Engineering

University of Newcastle-upon-Tyne

ELSEVIER

BUTERWORTH

H E M E M A "

AMSTERDAM BOSTON HEIDELBERG 0 LONDON NEWYORK OXFORD

PARIS SAN DlEGO SAN FRANCISCO* SINGAPORE SYDNEY TOKYO

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Copyright 0 1990, 1996, 1999, J H Harker and J R Backhurst, J M Coulson,

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Preface to Sixth Edition

It is somewhat sobering to realise that the sixth edition of Volume 1 appears 45 years

after the publication of the first edition in 1954 Over the intervening period, there have

been considerable advances in both the underlying theory and the practical applications

of Chemical Engineering; all of which are reflected in parallel developments in under- graduate courses In successive editions, we have attempted to adapt the scope and depth

of treatment in the text to meet the changes in the needs of both students and practitioners

of the subject

Volume 1 continues to concentrate on the basic processes of Momentum Transfer (as

in fluid flow), Heat Transfer, and Mass Transfer, and it is also includes examples of practical applications of these topics in areas of commercial interest such as the pumping of fluids, the design of shell and tube heat exchangers and the operation and performance

of cooling towers In response to the many requests from readers (and the occasional note of encouragement from our reviewers), additional examples and their solutions have now been included in the main text The principal areas of application, particularly of the

theories of Mass Transfer across a phase boundary, form the core material of Volume 2

however, whilst in Volume 6, material presented in other volumes is utilised in the practical design of process plant

The more important additions and modifications which have been introduced into this sixth edition of Volume 1 are:

Dimensionless Analysis The idea and advantages of treating length as a vector quantity

and of distinguishing between the separate role of mass in representing a quantity of matter

as opposed to its inertia are introduced

Fluid Flow The treatment of the behaviour of non-Newtonian fluids is extended and

the methods used for pumping and metering of such fluids are updated

Heat Transfer A more detailed discussion of the problem of unsteady-state heat transfer

by conduction where bodies of various shapes are heated or cooled is offered together with a more complete treatment of heat transfer by radiation and a re-orientation of the introduction to the design of shell and tube heat exchangers

Mass Transfer The section on mass transfer accompanied by chemical reaction has been considerably expanded and it is hoped that this will provide a good basis for the understanding of the operation of both homogeneous and heterogeneous catalytic reac- tions

As ever, we are grateful for a great deal of help in the preparation of this new edition from a number of people In particular, we should like to thank Dr D.G Peacock for the great enthusiasm and dedication he has shown in the production of the Index, a task he has

undertaken for us over many years We would also mention especially Dr R.P Chhabra

of the Indian Institute of Technology at Kanpur for his contribution on unsteady-state heat transfer by conduction, those commercial organisations which have so generously contributed new figures and diagrams of equipment, our publishers who cope with our

xv

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mi CHEMICAL ENGINEERING

perhaps overwhelming number of suggestions and alterations with a never-failing patience and, most of all, our readers who with great kindness, make so many extremely useful and helpful suggestions all of which, are incorporated wherever practicable With their continued help and support, the signs are that this present work will continue to be of real value as we move into the new Millenium

Swansea, I999

Newcastle upon Tyne, 1999

J.F RICHARDSON

J.R BACKHURST J.H HARKER

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Contents

Professor J M Coulson

Preface to Sixth Edition

Preface to F i f h Edition

Preface to Fourth Edition

Preface to Thud Edition

Preface to Second Edition

Preface to First Edition

1.2.3 The foot-pound-second (fps) system

1.2.4 The British engineering system

Redefinition of the length and mass dimensions

1.6.1 Vector and scalar quantities

1.6.2 Quantity mass and inertia mass

Further reading

References

Nomenclature

Pressure signal transmission- the differential pressure cell

Measurement by flow through a constriction

Pressure recovery in orifice-type meters Other methods of measuring flowrates

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viii CONTENTS

7 Liquid Mixing

7.1 Introduction - types of mixing

7.1.1 Single-phase liquid mixing

7.1.2 Mixing of immiscible liquids

7.3 Scale-up of stirred vessels

7.4 Power consumption in stirred vessels

7.4.1 Low viscosity systems

7.4.2 High viscosity systems

Mow patterns in stirred tanks

Rate and time for mixing

7.7.1 Mechanical agitation

7.7.2 Portable mixers

7.7.3 Extruders

7.7.4 Static mixers

7.7.5 Other types of mixer

7.8 Mixing in continuous systems

8.2.2 Positive-displacement rotary pumps

8.2.3 The centrifugal pump

8.3 Pumping equipment for gases

8.3.1 Fans and rotary compressors

8.3.2 Centrifugal and turbocompressors

8.3.3 The reciprocating piston compressor

8.3.4

8.4 The use of compressed air for pumping

8.4 I The air-lift pump

Power required for the compression of gases

Part 2 Heat Transfer

9 Heat Transfer

9.1 Introduction

9.2 Basic considerations

9.2 I

9.2.2 Mean temperature difference

Individual and overall coefficients of heat transfer

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Heat transfer by convection

9.4.1 Natural and forced convection

9.4.2

9.4.3 Forced convection in tubes

9.4.4 Forced convection outside tubes

9.4.5 Flow in non-circular sections

9.4.6 Convection to spherical particles

9.4.7 Natural convection

Heat transfer by radiation

9.5.1 Introduction

9.5.2 Radiation from a black body

9.5.3 Radiation from real surfaces

9.5.4 Radiation transfer between black surfaces

9.5.5 Radiation transfer between grey surfaces

9.5.6 Radiation from gases

Heat transfer in the condensation of vapours

Heat transfer in reaction vessels

9.8.1 Helical cooling coils

9.9.6 Heat exchanger design

9.9.7 Heat exchanger performance

Conduction through a plane wall

Conduction through a thick-walled tube

Conduction through a spherical shell and to a particle

Conduction with internal heat source

Application of dimensional analysis to convection

Film coefficients for vertical and inclined surfaces

Condensation on vertical and horizontal tubes

Heat transfer coefficients and heat Bux

Analysis based on bubble characteristics

Time required for heating or cooling

Mean temperature difference in multipass exchangers

Pressure drop in heat exchangers

9.9.8 Transfer units

9.10 Other forms of equipment

9.10.1 Finned-tube units

9.10.2 Plate-type exchangers

9.10.3 Spiral heat exchangers

9.10.4 Compact heat exchangers

9.10.5 Scraped-surface heat exchangers

9.1 1.1 Heat losses through lagging

9.1 1.2 Economic thickness of lagging

9.1 1.3 Critical thickness of lagging

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Diffusion in binary gas mixtures

10.2 I Properties of binary mixtures

10.2.2 Equimolecular counterdiffusion

10.2.3 Mass transfer through a stationary second component

10.2.4 Diffusivities of gases and vapours

10.2.5 Mass transfer velocities

10.2.6 General case for gas-phase mass transfer in a binary mixture

10.2.7 Diffusion as a mass flux

10.2.8 Thermal diffusion

10.2.9 Unsteady-state mass transfer

Multicomponent gas-phase systems

10.3.1

10.3.2 Maxwell’s law of diffusion

Diffusion in liquids

10.4.1 Liquid phase diffusivities

Mass transfer across a phase boundary

10.5.1 The two-film theory

10.5.2 The penetration theory

10.5.3 The film-penetration theory

10.5.4 Mass transfer to a sphere in a homogenous fluid

10.5.5 Other theories of mass transfer

10.7.3 Other particle shapes

10.7.4 Mass transfer and chemical reaction with a mass transfer resistance

external to the pellet Practical studies of mass transfer

10.8.1 The j-factor of Chilton and Colburn for flow in tubes

10.8.2 Mass transfer at plane surfaces

10.8.3 Effect of surface roughness and form drag

10.8.4 Mass transfer from a fluid to the surface of particles

Molar flux in terms of effective diffusivity

Countercurrent mass transfer and transfer units

10.9 Further reading

10.10 References

10.1 1 Nomenclature

Part 4 Momentum, Heat and Mass Transfer

11 The Boundary Layer

11.1 Introduction

1 1.2 The momentum equation

11.3 The streamline portion of the boundary layer

11.4 The turbulent boundary layer

11.4.1 The turbulent portion

11.4.2 The laminar sub-layer

Boundary layer theory applied to pipe flow

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Universal velocity profile

12.4.1 The turbulent core

12.4.2 The laminar sub-layer

12.4.3 The buffer layer

12.4.4 Velocity profile for all regions

12.4.5 Velocity gradients

12.4.6 Laminar sub-layer and buffer layer thicknesses

12.4.7 Variation of eddy kinematic viscosity

12.4.8 Approximate form of velocity profile in turbulent region

12.4.9 Effect of curvature of pipe wall on shear stress

Friction factor for a smooth pipe

Effect of surface roughness on shear stress

Simultaneous momentum, heat and mass transfer

Reynolds analogy

12.8.1 Simple form of analogy between momentum, heat and mass transfer

12.8.2 Mass transfer with bulk flow

12.8.3 Taylor-Prandtl modification of Reynolds analogy for heat

transfer and mass transfer 12.8.4 Use of universal velocity profile in Reynolds analogy

12.8.5 Flow over a plane surface

13.2.3 Adiabatic saturation temperature

Humidity data for the air-water system

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Humidification and dehumidification

13.5.1 Methods of increasing humidity

13.5.2 Dehumidification

Water cooling

13.6.1 Cooling towers

13.6.2 Design of natural-draught towers

13.6.3 Height of packing for both natural and mechanical draught towers

13.6.4 Change in air condition

13.6.5 Temperature and humidity gradients in a water cooling tower

13.6.6 Evaluation of heat and m a s transfer coefficients

A I Tables of physical properties

A2 Steam tables

A3 Mathematical tables

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introduction of the Systkme International d’llnith (SI)(1,2) to be discussed later, which is used throughout all the Volumes of this series of books This system is now in general use

in Europe and is rapidly being adopted throughout the rest of the world, including the USA

where the initial inertia is now being overcome Most of the physical properties determined

in the laboratory will originally have been expressed in the cgs system, whereas the dimensions of the full-scale plant, its throughput, design, and operating characteristics appear either in some form of general engineering units or in special units which have their origin in the history of the particular industry This inconsistency is quite unavoidable and is a reflection of the fact that chemical engineering has in many cases developed as

a synthesis of scientific knowledge and practical experience Familiarity with the various systems of units and an ability to convert from one to another are therefore essential,

as it will frequently be necessary to access literature in which the SI system has not been used In this chapter the main systems of units are discussed, and the importance of understanding dimensions emphasised It is shown how dimensions can be used to help very considerably in the formulation of relationships between large numbers of parameters The magnitude of any physical quantity is expressed as the product of two quantities; one is the magnitude of the unit and the other is the number of those units Thus the distance between two points may be expressed as 1 m or as 100 cm or as 3.28 ft The metre, centimetre, and foot are respectively the size of the units, and 1, 100, and 3.28 are the corresponding numbers of units

Since the physical properties of a system are interconnected by a series of mechanical and physical laws, it is convenient to regard certain quantities as basic and other quantities

as derived The choice of basic dimensions varies from one system to another although

it is usual to take length and time as fundamental These quantities are denoted by L and

T The dimensions of velocity, which is a rate of increase of distance with time, may be written as LT-’, and those of acceleration, the rate of increase of velocity, are LT-2 An area has dimensions L2 and a volume has the dimensions L3

The volume of a body does not completely define the amount of material which it contains, and therefore it is usual to define a third basic quantity, the amount of matter in the body, that is its mass M Thus the density of the material, its mass per unit volume, has the dimensions ML-3 However, in the British Engineering System (Section 1.2.4) force F is used as the third fundamental and mass then becomes a derived dimension

1

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The proportionality constant therefore has the dimensions:

In any set of consistent or coherent units the proportionality constant in equation 1.1 is

put equal to unity, and unit force is that force which will impart unit acceleration to unit mass Provided that no other relationship between force and mass is used, the constant may be arbitrarily regarded as dimensionless and the dimensional relationship:

is obtained

If, however, some other physical law were to be introduced so that, for instance, the attractive force between two bodies would be proportional to the product of their masses, then this relation between F and M would no longer hold It should be noted that mass has essentially two connotations First, it is a measure of the amount of material and appears

in this role when the density of a fluid or solid is considered Second, it is a measure of the inertia of the material when used, for example, in equations 1.1-1.3 Although mass

is taken normally taken as the third fundamental quantity, as already mentioned, in some engineering systems force is used in place of mass which then becomes a derived unit

the m k s system which employs larger units of mass and length (kilogram in place of gram,

and metre in place of centimetre); this system has been favoured by electrical engineers because the fundamental and the practical electrical units (volt, ampere and ohm) are then

identical The SI system is essentially based on the mks system of units

1.2.1 The centimetre-gram-second (cgs) system

In this system the basic units are of length L, mass M, and time T with the nomenclature:

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Table 1.1

Units

Mass gram kilogram pound M slug FL-'T2 M

Pressure dyndsquare centimetre Newtonlsq metre poundaUsquare foot m-1 T-2 p o u n d f d s q u a r e f o o t FL-2 cn

unitmass erg/gram"C ~o~ie/kiiogram K foot-pomdal/pound "c L ~ T - ~ O - ' foot-pound/slug "F L2T-'B'

Universal gas 8.314 x lo7 ergbole "C 8314 Jkmol K 8.94 fi-poundaMb mol "C MN-'L2T-20-' 4.96 x 104 foot-pound

Mechanical equivalent of heat, J 4.18 x lo7 erg/gram-"C 1 J (heat energy) = 1 J 2.50 x 104 foot-pounWpound "F L2T-'8-' H- I ML2 T-2

(mechanical energy)

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4 CHEMICAL ENGINEERING

The unit of force is that force which will give a mass of 1 g an acceleration of 1 cm/s2 and is known as the dyne:

1.2.2 The metre-kilogram-second (mks system and the Systeme

international d'Unites (SI)

These systems are in essence modifications of the cgs system but employ larger units, The basic dimensions are again of L, M, and T

The unit of force, known as the Newton, is that force which will give an acceleration

of 1 m/s2 to a mass of one kilogram Thus 1 N = 1 kg m / s 2 with dimensions MLT-2,

and one Newton equals 1 6 dynes The energy unit, the Newton-metre, is lo7 ergs and is called the Joule; and the power unit, equal to one Joule per second, is known as the Watt

For many purposes, the chosen unit in the SI system will be either too large or too small for practical purposes, and the following prefixes are adopted as standard Multiples

or sub-multiples in powers of lo3 are preferred and thus, for example, millimetre should always be used in preference to centimetre

lo-'

10-2

10-3 10-6 10-9 10-12 10-15 10-18

deci centi milli micro nano pic0 femto alto These prefixes should be used with great care and be written immediately adjacent to the unit to be qualified; furthermore only one prefix should be used at a time to precede

a given unit Thus, for example, metre, which is one millimetre, is written 1 mm lo3 kg is written as 1 Mg, not as 1 kkg This shows immediately that the name kilogram

is an unsuitable one for the basic unit of mass and a new name may well be given to it

in the future

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UNITS AND DIMENSIONS 5

Some special terms are acceptable, and commonly used in the SI system and, for example, a mass of lo3 kg (1 Mg) is called a tonne (t); and a pressure of 100 kN/m2 is called a bar

The most important practical difference between the m k s and the SI systems lies in the

units used for thermal energy (heat), and this topic is discussed in Secton 1.2.7

A detailed account of the structure and implementation of the SI system is given in

a publications of the British Standards Institution('), and of Her Majesty's Stationery Office(2)

1.2.3 The foot-pound-second (fps) system

The basic units in this system are:

The unit of force gives that which a mass of 1 lb an acceleration of 1 ft/s2 is known The unit of energy (or work) is the foot-poundal, and the unit of power is the foot-

as the poundal (pdl)

poundal per second

1.2.4 The British engineering system

In an alternative form of the fps system (Engineering system) the units of length (ft) and time (s) are unchanged, but the third fundamental is a unit of force (F) instead of mass and is known as the pound force (lbf) This is defined as the force which gives a mass

of 1 lb an acceleration of 32.1740 ft/s2, the "standard" value of the acceleration due to gravity It is therefore a fixed quantity and must not be confused with the pound weight which is the force exerted by the earth's gravitational field on a mass of one pound and which varies from place to place as g varies, It will be noted therefore that the pound force and the pound weight have the same value only when g is 32.1740 ft2/s

The unit of mass in this system is known as the slug, and is the mass which is given

an acceleration of 1 ft/s2 by a one pound force:

1 Slug = 1 Ibf ft-'S2

Misunderstanding often arises from the fact that the pound which is the unit of mass

in the fps system has the same name as the unit of force in the engineering system To avoid confusion the pound mass should be written as lb or even lb, and the unit of force always as lbf

It will be noted that:

1 slug = 32.1740 lb mass and 1 lbf = 32.1740 pdl

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6 CHEMICAL ENGINEERING

To summarise:

The basic units are:

The derived units are:

Note: 1 horsepower is defined as 550 ft-lbf/s

1.2.5 Non-coherent system employing pound mass and pound force simultaneously

TWO units which have never been popular in the last two systems of units (Sections 1.2.3 and 1.2.4) are the poundal (for force) and the slug (for mass) As a result, many writers, particularly in America, use both the pound mass and pound force as basic units in the same equation because they are the units in common use This is an essentially incoherent system and requires great care in its use In this system a proportionality factor between force and mass is defined as g , given by:

Force (in pounds force) =(mass in pounds) (acceleration in ft/s2)/gc

From equation 1.4, it is seen that g, has the dimensions F-'MLTV2 or, putting F = MLT-2,

it is seen to be dimensionless Thus:

g , = 32.1740 lbf/(lb,ft s-~)

1 ft K2

gc = i.e g , is a dimensionless quantity whose numerical value corresponds to the acceleration due to gravity expressed in the appropriate units

(It should be noted that a force in the cgs system is sometimes expressed as a g r a m

force and in the mks system as kilogram force, although this is not good practice It should also be noted that the gram force = 980.665 dyne and the kilogram force = 9.80665 N)

1.2.6 Derived units

The three fundamental units of the SI and of the cgs systems are length, mass, and time It has been shown that force can be regarded as having the dimensions of MLT-2, and the dimensions of many other parameters may be worked out in terms of the basic MLT system

energy is given by the product of force and distance with dimensions ML2T-2, and pressure is the force per unit area with dimensions ML-1T-2

For example:

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viscosity is defined as the shear stress per unit velocity gradient with dimensions and kinematic viscosity is the viscosity divided by the density with dimensions The units, dimensions, and normal form of expression for these quantities in the SI

1.2.7 Thermal (heat) units

Heat is a form of energy and therefore its dimensions are ML2T-2 In many cases, however, no account is taken of interconversion of heat and "mechanical" energy (for example, kinetic, potential and kinetic energy), and heat can treated as a quantity which

is conserved It may then be regarded as having its own independent dimension H which can be used as an additional fundamental It will be seen in Section 1.4 on dimensional analysis that increasing the number of fundamentals by one leads to an additional relation and consequently to one less dimensionless group

Wherever heat is involved temperature also fulfils an important role: firstly because the heat content of a body is a function of its temperature and, secondly, because temperature difference or temperature gradient determines the rate at which heat is transferred Temperature has the dimension 8 which is independent of M,L and T, provided that no resort is made to the kinetic theory of gases in which temperature is shown to be directly proportional to the square of the velocity of the molecules

It is not incorrect to express heat and temperature in terms of the M,L,T dimensions, although it is unhelpful in that it prevents the maximum of information being extracted from the process of dimensional analysis and reduces the insight that it affords into the physical nature of the process under consideration

Dimensionally, the relation between H, M and 8 can be expressed in the form:

where C, the specific heat capacity has dimensions H M-'B-'

Equation 1.5 is similar in nature to the relationship between force mass and accelara- tion given by equation 1.1 with one important exception The proportionality constant in equation 1.1 is not a function of the material concerned and it has been possible arbitrarily

to put it equal to unity The constant in equation 1.5, the specific heat capacity C,, differs from one material to another

In the SI system, the unit of heat is taken as the same as that of mechanical energy and is therefore the Joule For water at 298 K (the datum used for many definitions), the specific heat capacity C, is 4186.8 J/kg K

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Prior to the now almost universal adoption of the SI system of units, the unit of heat was defined as the quantity of heat required to raise the temperature of unit mass of water by one degree This heat quantity is designated the calorie in the cgs system and

the kilocalorie in the m k s system, and in both cases temperature is expressed in degrees

Celsius (Centigrade) As the specific heat capacity is a function of temperature, it has been necessary to set a datum temperature which is chosen as 298 K or 25°C

In the British systems of units, the pound, but never the slug, is taken as the unit

of mass and temperature may be expressed either in degrees Centigrade or in degrees

Fahrenheit The units of heat are then, respectively, the pound-calorie and the British thermal unit (Btu) Where the Btu is too small for a given application, the therm (= lo5

Btu) is normally used

Thus the following definitions of heat quantities therefore apply:

scale (degrees)

Centigrade heat unit (CHU)

1 CHU = 1.8 Btu

In all of these systems, by definition, the specific heat capacity of water is unity It may

be noted that, by comparing the definitions used in the SI and the mks systems, the kilocalorie is equivalent to 4186.8 J k g K This quantity has often been referred to as the

mechanical equivalent of heat J

1.2.8 Molar units

When working with ideal gases and systems in which a chemical reaction is taking place, it

is usual to work in terms of molar units rather than mass The mole (mol) is defined in the

SI system as the quantity of material which contains as many entities (atoms, molecules or formula units) as there are in 12 g of carbon 12 It is more convenient, however, to work

in terms of the kilomole (kmol) which relates to 12 kg of carbon 12, and the kilomole

is used exclusively in this book The number of molar units is denoted by dimensional symbol N The number of kilomoles of a substance A is obtained by dividing its mass in kilograms (M) by its molecular weight MA M A thus has the dimensions MN-' The Royal

Society recommends the use of the term relative molecular mass in place of molecular weight, but molecular weight is normally used here because of its general adoption in the

processing industries

1.2.9 Electrical units

Electrical current (I) has been chosen as the basic SI unit in terms of which all other

electrical quantities are defined Unit current, the ampere (A, or amp), is defined in

terms of the force exerted between two parallel conductors in which a current of 1 amp is flowing Since the unit of power, the watt, is the product of current and potential difference,

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the volt (V) is defined as watts per amp and therefore has dimensions of M L ~ T - ~ I - '

From Ohm's law the unit of resistance, the ohm, is given by the ratio volts/amps and therefore has dimensions of ML2T-31-2 A similar procedure may be followed for the evaluation of the dimensions of other electrical units

Conversion of units from one system to another is simply carried out if the quantities are expressed in terms of the fundamental units of mass, length, time, temperature Typical conversion factors for the British and metric systems are:

Mass 1 lb = ( - 3im2) slug = 453.6 g = 0.4536 kg

Length 1 ft = 30.48 cm = 0.3048 m

Temperature 1°F = (A) "C = (A) K (or deg.K)

difference

Force 1 pound force = 32.2 poundal = 4.44 x lo5 dyne = 4.44 N

Other conversions are now illustrated

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UNITS AND DIMENSIONS

: 2.989'1 kN/mz

: 249.09 N/m2

: 3.386,4 kN/mZ : 133.32 N/mz

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Table 1.2 (continued)

Surface energy

Mass flux density

Heat flux density

Heat transfer

coefficient

Specific enthalpy

(latent heat, etc.)

Specific heat capacity

1 Btuh ft2"F '1 Btdlb

Dimensional analysis depends upon the fundamental principle that any equation or relation

between variables must be dimensionally consistent; that is, each term in the relationship

must have the same dimensions Thus, in the simple application of the principle, an equation may consist of a number of terms, each representing, and therefore having, the dimensions of length It is not permissible to add, say, lengths and velocities in an algebraic equation because they are quantities of different characters The corollary of this principle is that if the whole equation is divided through by any one of the terms, each

remaining term in the equation must be dimensionless The use of these dimensionless

groups, or dimensionless numbers as they are called, is of considerable value in developing relationships in chemical engineering

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of

the technique of dimensional analysis which enables the variables in a problem to be

grouped into the form of dimensionless groups Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants

The study of problems in fluid dynamics and in heat transfer is made difficult by the many parameters which appear to affect them In most instances further study shows that the variables may be grouped together in dimensionless groups, thus reducing the effective

number of variables It is rarely possible, and certainly time consuming, to try to vary

these many variables separately, and the method of dimensional analysis in providing a smaller number of independent groups is most helpful to the investigated

The application of the principles of dimensional analysis may best be understood by considering an example

It is found, as a result of experiment, that the pressure difference (AP) between two ends of a pipe in which a fluid is flowing is a function of the pipe diameter d , the pipe

length 1 , the fluid velocity u, the fluid density p, and the fluid viscosity p

The relationship between these variables may be written as:

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The form of the function is unknown, though since any function can be expanded as

a power series, the function may be regarded as the sum of a number of terms each consisting of products of powers of the variables The simplest form of relation will be where the function consists simply of a single term, or:

The conditions of dimensional consistency must be met for each of the fundamentals

of M, L, and T and the indices of each of these variables may be equated Thus:

n4 = 1 - n5 (from the equation in M)

ng = 2 - n5 (from the equation in T)

Substituting in the equation for L:

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The group u d p l p , known as the Reynolds number, is one which frequently arises in

the study of fluid flow and affords a criterion by which the type of flow in a given

geometry may be characterised Equation 1.8 involves the reciprocal of the Reynolds number, although this may be rewritten as:

- AP = const (6)”’ ( 3 - n 5 PU2

of this chapter, this statement will be generalised to show that the number of dimensionless

groups is normally the number of variables less the number of fundamentals (but see the

note in Section 1.5)

A number of important points emerge from a consideration of the preceding example:

1 If the index of a particular variable is found to be zero, this indicates that this variable

is of no significance in the problem

2 If two of the fundamentals always appear in the same combination, such as L and

T always occuring as powers of LT-’, for example, then the same equation for the indices will be obtained for both L and T and the number of effective fundamentals

is thus reduced by one

3 The form of the final solution will depend upon the method of solution of the simultaneous equations If the equations had been solved, say, in terms of n3 and n4

instead of 122 and n5, the resulting dimensionless groups would have been different, although these new groups would simply have been products of powers of the original groups Any number of fresh groups can be formed in this way

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume

The choice of physical variables to be included in the dimensional analysis must be based on an understanding of the nature of the phenomenon being studied although, on occasions there may be some doubt as to whether a particular quantity is relevant or not

If a variable is included which does not exert a significant influence on the problem, the value of the dimensionless group in which it appears will have little effect on the final numerical solution of the problem, and therefore the exponent of that group must approach zero This presupposes that the dimensionless groups are so constituted that the variable in

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question appears in only one of them On the other hand if an important variable is omitted,

it may be found that there is no unique relationship between the dimensionless groups Chemical engineering analysis requires the formulation of relationships which will apply over a wide range of size of the individual items of a plant This problem of scale

up is vital and it is much helped by dimensional analysis

Since linear size is included among the variables, the influence of scale, which may be regarded as the influence of linear size without change of shape or other variables, has been introduced Thus in viscous flow past an object, a change in linear dimension L will alter the Reynolds number and therefore the flow pattern around the solid, though if the change in scale is accompanied by a change in any other variable in such a way that the Reynolds number remains unchanged, then the flow pattern around the solid will not be altered This ability to change scale and still maintain a design relationship is one of the many attractions of dimensional analysis

It should be noted that it is permissible to take a function only of a dimensionless quantity It is easy to appreciate this argument when account is taken of the fact that any function may be expanded as a power series, each term of which must have the same dimensions, and the requirement of dimensional consistency can be met only if these terms and the function are dimensionless Where this principle appears to have been invalidated, it is generally because the equation includes a further term, such as an integration constant, which will restore the requirements of dimensional consistency For example, l: * = Inn - lnx,, and if x is not dimensionless, it appears at first sight that

X

the principli has been infringed Combining the two logarithmic terms, however, yields

X

In (:) , and X, is clearly dimensionless In the case of the indefinte integral, lnx, would,

in eff&, have been the integration constant

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each

of the fundamentals A generalisation of this statement is provided in Buckingham’s n

theorem(4) which states that the number of dimensionless groups is equal to the number

of variables minus the number of fundamental dimensions In mathematical terms, this can be expressed as follows:

If there are n variables, Ql, Q2, , Qn, the functional relationship between them may

be written as:

If there are m fundamental dimensions, there will be (n - m) dimensionless groups

( H I , n2, , n,-,) and the functional relationship between them may be written as:

The groups nl, n2, and so on must be independent of one another, and no one group should be capable of being formed by multiplying together powers of the other groups

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By making use of this theorem it is possible to obtain the dimensionless groups more simply than by solving the simultaneous equations for the indices Furthermore, the functional relationship can often be obtained in a form which is of more immediate use

The method involves choosing m of the original variables to form what is called a recurring set Any set m of the variables may be chosen with the following two provisions:

(1) Each of the fundamentals must appear in at least one of the m variables

(2) It must not be possible to form a dimensionless group from some or all of the variables within the recurring set If it were so possible, this dimensionless group would, of course, be one of the ll terms Thus, the number of dimensionless groups

is increased by one for each of the independent groups that can be so formed The procedure is then to take each of the remaining ( n - m ) variables on its own and

to form it into a dimensionless group by combining it with one or more members of the

recurring set In this way the (n - m) ll groups are formed, the only variables appearing

in more than one group being those that constitute the recumng set Thus, if it is desired

to obtain an explicit functional relation for one particular variable, that variable should not be included in the recurring set

In some cases, the number of dimensionless groups will be greater than predicted

by the ll theorem For instance, if two of the fundamentals always occur in the same combination, length and time always as LT-*, for example, they will constitute a single

fundamental instead of two fundamentals By referring back to the method of equating indices, it is seen that each of the two fundamentals gives the same equation, and therefore only a single constraint is placed on the relationship by considering the two variables

Thus, although m is normally the number of fundamentals, it is more strictly defined as the maximum number of variables from which a dimensionless group cannot be formed

The procedure is more readily understood by consideration of the illustration given previously The relationship between the variables affecting the pressure drop for flow of fluid in a pipe may be written as:

(1) Both 1 and d cannot be chosen as they can be formed into the dimensionless group

(2) AP, p and u cannot be used since AP/pu* is dimensionless

l l d

Outside these constraints, any three variables can be chosen It should be remembered, however, that the variables forming the recurring set are liable to appear in all the dimensionless groups As this problem deals with the effect of conditions on the pressure difference AP, it is convenient if A P appears in only one group, and therefore it is preferable not to include it in the recurring set

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If the variables d , u, p are chosen as the recurring set, this fulfils all the above conditions Dimensionally:

Group l l l is, therefore,

A glass particle settles under the action of gravity in a liquid Obtain a dimensionless grouping of the variables

involved The falling velocity is found to be proportional to the square of the particle diameter when the other variables are constant What will be the effect of doubling the viscosity of the liquid?

Solution

It may be expected that the variables expected to influence the terminal velocity of a glass particle settling in

a liquid, are:

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Table 1.3 Some important dimensionless groups

Symbol Name of group In terms of Definition Application

Flow of viscoelastic fluid

Pressure and momentum in fluid

Unsteady state heat transfer/mass transfer

Fluid flow with free surface

Heat transfer by natural convection

- "' Heat transfer to fluid in tube

Gas flow at high velocity

Heat transfer in fluid

Fluid flow and heat transfer

Fluid flow and mass transfer

Heat transfer in flowing fluid

Fluid flow involving viscous and inertial forces

Mass transfer in flowing fluid

- 'Dl Mass transfer in fluid

- Fluid flow with intertacial forces

Fluid drag at surface

Power consumption for mixers

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particle diameter d; particle density, ps; liquid density, p; liquid viscosity, y and the acceleration due to gravity, g

Particle density ps is important because it determines the gravitational (accelerating) force on the particle However when immersed in a liquid the particle receives an upthrust which is proportional to the liquid density

p The effective density of the particles (ps - p ) is therefore used in this analysis Then:

w = f(d, ( ~ s - P), P, ~9 8) The dimensions of each variable are:

uo = LT-', d = L, ps - p = ML-3, p = ML-3,

y = ML-IT-' and g = LT-2

With six variables and three fundamental dimensions, (6 - 3) = 3 dimensionless groups are expected Choosing

d, p and y as the recurring set:

The rate at which a drop spreads, say UR m/s, will be influenced by:

viscosity of the liquid, p-dimensions = ML-lT-l

volume of the drop, V-dimensions = L3

density of the liquid, p-dimensions = ML-3

acceleration due to gravity, g-dimensions = LT-'

and possibly, surface tension of the liquid, a-dimensions = MT-z

Taking V, p and g as the recurring set:

Noting the dimensions of UR as LT-', there are six variables and hence (6 - 3) = 3 dimensionless groups

V L3 and L = v0.33

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1.6.1 Vector and scalar quantities

It is important to recognise the differences between scalar quantities which have a magnitude but no direction, and vector quantities which have both magnitude and direction Most length terms are vectors in the Cartesian system and may have components in the X, Y

and Z directions which may be expressed as Lx, Ly and Lz There must be dimensional consistency in all equations and relationships between physical quantities, and there is therefore the possibility of using all three length dimensions as fundamentals in dimensional analysis This means that the number of dimensionless groups which are formed will be less

Combinations of length dimensions in areas, such as LxLy, and velocities, accelerations

and forces are all vector quantities On the other hand, mass, volume and heat are all

scalar quantities with no directional significance The power of dimensional analysis is

thus increased as a result of the larger number of fundamentals which are available for use Furthermore, by expressing the length dimension as a vector quantity, it is possible

to obviate the difficulty of two quite different quantities having the same dimensions For example, the units of work or energy may be obtained by multiplying a force in the X-direction (say) by a distance also in the X-direction The dimensions of energy are therefore:

2 -2 (MLxT-2)(Lx) = MLxT

It should be noted in this respect that a torque is obtained as a product of a force in the

X-direction and an arm of length Ly, say, in a direction at right-angles to the Y-direction Thus, the dimensions of torque are MLxLYT-~, which distinguish it from energy Another benefit arising from the use of vector lengths is the ability to differentiate

between the dimensions of frequency and angular velocity, both of which are T-' if length is treated as a scalar quantity Although an angle is dimensionless in the sense that

it can be defined by the ratio of two lengths, its dimensions become Lx/Ly if these two

lengths are treated as vectors Thus angular velocity then has the dimensions LxG'T-' compared with T-' for frequency

Of particular interest in fluid flow is the distinction between shear stress and pressure

(or pressure difference), both of which are defined as force per unit area For steady-state

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UNITS AND DIMENSIONS 21 flow of a fluid in a pipe, the forces attributable to the pressure difference and the shear stress must balance The pressure difference acts in the axial X-direction, say, and the area

A on which it acts lies in the Y-2 plane and its dimensions can therefore be expressed

as LyLz On the other hand, the shear stress R which is also exerted in the X-direction

acts on the curved surface of the walls whose area S has the dimensions LXLR where LR

is length in the radial direction Because there is axial symetry, LR can be expressed as

L:”Li’’ and the dimensions of S are then LxL~’’L~”

The force F acting on the fluid in the X (axial)-direction has dimensions MLxT-’,

and hence:

and R = F/S has dimensions MLxT”/LxL:/’Li’’ = MLyl/’L-’/’T-’ z

giving dimensions of A P / R as LxLY’/’L,’/’ or LxL,’ (which would have been dimensionless had lengths not been treated as vectors)

For a pipe of radius r and length I, the dimensions of r/1 are &‘LR and hence ( A P I R ) ( r / l ) is a dimensionless quantity The role of the ratio r/l would not have been established had the lengths not been treated as vectors It is seen in Chapter 3 that this conclusion is consistent with the results obtained there by taking a force balance on the fluid

1.6.2 Quantity mass and inertia mass

The term mass M is used to denote two distinct and different properties:

1 The quantity of matter M,, and

2 The inertial property of the matter Mi

These two quantities are proportional to one another and may be numerically equal, although they are essentially different in kind and are therefore not identical The distinction is particularly useful when considering the energy of a body or of a fluid

Because inertial mass is involved in mechanical energy, the dimensions of all energy terms are MiL’T-’ Inertial mass, however, is not involved in thermal energy (heat) and therefore specific heat capacity C, has the dimensions MiL2T2/M,B = MiM;’ L2T-’8-’

or HM;’B-’ according to whether energy is expressed in, joules or kilocalories, for example

In practical terms, this can lead to the possibility of using both mass dimensions as fundamentals, thereby achieving similar advantages to those arising from taking length

as a vector quantity This subject is discussed in more detail by HUNTLEY(’)

WARNING

Dimensional analysis is a very powerful tool in the analysis of problems involving a large number of variables However, there are many pitfalls for the unwary, and the technique should never be used without a thorough understanding of the underlying basic principles

of the physical problem which is being analysed

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ASTAFUTA, G.: Chem Eng Sci 52 (1997) 4681 Dimensional analysis, scaling and orders of magnitude BLACKMAN, D R.: SI Units in Engineering (Macmillan, 1969)

BRIDCMAN, P W.: Dimensional Analysis (Yale University Press, 1931)

FOCKEN, C M.: Dimensional Methods and their Applications (Edward Arnold, London, 1953)

I-kwrrr, G F.: Proc I l t h International Heat Transfer Conference Kyongju, Korea (1998) 1

HUNTLEY, H E.: Dimensional Analysis (Macdonald and Co (Publishers) Ltd, London, 1952)

IPSEN, D C.: Units, Dimensions, and Dimensionless Numbers (McGraw-Hill, 1960)

JOHNSTONE, R E and THRING, M W.: Pilot Plants, Models and Scale-up in Chemical Engineering (McGraw-

KLINKENBERG, A and MOOY, H H.: Chem Eng Prog 44 (1948) 17 Dimensionless groups in fluid friction, MASSEY, B S.: Units, Dimensional Analysis and Physical Similarity (VAN Nostrand Reinhold, 1971)

MULLIN, J W.: The Chemical Engineer (London) No 21 1 (Sept 1967) 176 SI units in chemical engineering MULLIN, J W.: The Chemical Engineer (London) No 254 (1971) 352 Recent developments in the change-over British Standards Institution Publication PD 5686 (1967) The Use of SI Units

Quantities, Units and Symbols (The Symbols Committee of the Royal Society, 1971)

SI The International System of Units (HMSO, 1970)

Hill, 1957)

heat and material transfer

to the International System of Units (SI)

1 British Standards Institution Publication PD 5686 (1967) The Use of SI Units

2 SI The International System of Units (HMSO, 1970)

3 MULLIN, J W The Chemical Engineer (London) No 21 1 (Sept 1967) 176, SI units in chemical engineering

4 BUCKINCHAM, E.: Phys Rev Ser., 2,4 (1914) 345 On physically similar systems: illustrations of the use of

5 HUNTLEY, H.E.: Dimensional Analysis (Madonald and Co (Publishers) Ltd, London,( 1952)

Mass rate of flow

Acceleration due to gravity

Numerical constant equal to standard

Heat transfer coefficient

Mass transfer coefficient

Electric current

Mechanical equivalent of heat

Thermal conductivity

Characteristic length or length of pipe

Molecular weight (relative molecular

L T - ~

-

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UNITS AND DIMENSIONS 23

Characteristic time for fluid

Characteristic time for process

N/m2 N/m2

Trang 36

PART 1

Fluid Flow

Trang 37

or, alternatively, the drag force exerted by the fluid on the particles In some cases, such

as filtration, the particles are in the form of a fairly stable bed and the fluid has to pass through the tortuous channels formed by the pore spaces In other cases the shape of the boundary surfaces must be so arranged that a particular flow pattern is obtained: for example, when solids are maintained in suspension in a liquid by means of agitation, the desired effect can be obtained with the minimum expenditure of energy as the most suitable flow pattern is produced in the fluid Further, in those processes where heat transfer or mass transfer to a flowing fluid occurs, the nature of the flow may have a profound effect on the transfer coefficient for the process

It is necessary to be able to calculate the energy and momentum of a fluid at various positions in a flow system It will be seen that energy occurs in a number of forms and that some of these are influenced by the motion of the fluid In the first part of this chapter the thermodynamic properties of fluids will be discussed It will then be seen how the thermodynamic relations are modified if the fluid is in motion In later chapters, the effects

of frictional forces will be considered, and the principal methods of measuring flow will

be described

When a fluid flows from one location to another, energy will, in general, be converted from one form to another The energy which is attributable to the physical state of the fluid

is known as internal energy; it is arbitrarily taken as zero at some reference state, such

as the absolute zero of temperature or the melting point of ice at atmospheric pressure

A change in the physical state of a fluid will, in general, cause an alteration in the internal energy An elementary reversible change results from an infinitesimal change in

27

Trang 38

one of the intensive factors acting on the system; the change proceeds at an infinitesimal rate and a small change in the intensive factor in the opposite direction would have caused the process to take place in the reverse direction Truly reversible changes never occur in practice but they provide a useful standard with which actual processes can be compared In an irreversible process, changes are caused by a finite difference in the intensive factor and take place at a finite rate In general the process will be accompanied

by the conversion of electrical or mechanical energy into heat, or by the reduction of the temperature difference between different parts of the system

For a stationary material the change in the internal energy is equal to the difference between the net amount of heat added to the system and the net amount of work done by the system on its surroundings For an infinitesimal change:

where dU is the small change in the internal energy, Sq the small amount of heat added, and SW the net amount of work done on the surroundings

In this expression consistent units must be used In the SI system each of the terms

in equation 2.1 is expressed in Joules per kilogram (Jkg) In other systems either heat units (e.g cal/g) or mechanical energy units (e.g erg/g) may be used dU is a small change in the internal energy which is a property of the system; it is therefore a perfect differential On the other hand, Sq and SW are small quantities of heat and work; they are not properties of the system and their values depend on the manner in which the change

is effected; they are, therefore, not perfect differentials For a reversible process, however, both Sq and SW can be expressed in terms of properties of the system For convenience, reference will be made to systems of unit mass and the effects on the surroundings will

be disregarded

A property called entropy is defined by the relation:

where dS is the small change in entropy resulting from the addition of a small quantity

of heat Sq, at a temperature T, under reversible conditions From the definition of the thermodynamic scale of temperature, $ Sq/T = 0 for a reversible cyclic process, and the net change in the entropy is also zero Thus, for a particular condition of the system, the entropy has a definite value and must be a property of the system; dS is, therefore, a perfect differential

For an irreversible process:

1: T d S = CSq+ CSF = q + F (say) (2.4)

When a process is isentropic, q = -F; a reversible process is isentropic when q = 0, that

is a reversible adiabatic process is isentropic

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FLOW OF FLUIDS-ENERGY AND MOMENTUM RELATIONSHIPS 29

The increase in the entropy of an irreversible process may be illustrated in the following manner Considering the spontaneous transfer of a quantity of heat Sq from one part of

a system at a temperature T1 to another part at a temperature T2, then the net change in

the entropy of the system as a whole is then:

T1 must be greater than T2 and dS is therefore positive If the process had been carried out reversibly, there would have been an infinitesimal difference between T I and T2 and the change in entropy would have been zero

The change in the internal energy may be expressed in terms of properties of the system itself For a reversible process:

Sq = T dS (from equation 2.2) and SW = P dv

if the only work done is that resulting from a change in volume, dv

Thus, from equation 2.1:

Since this relation is in terms of properties of the system, it must also apply to a system

in motion and to irreversible changes where the only work done is the result of change

of volume

Thus, in an irreversible process, for a stationary system:

from equations 2.1 and 2.2: dU = Sq - SW = T dS - P dv

As this relation is in terms of properties of the system, it must be applicable to all changes at constant volume

In an irreversible process:

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This quantity SF thus represents the mechanical energy which has been converted into heat and which is therefore available for increasing the temperature

For changes that take place under conditions of constant pressure, it is more satisfactory

to consider variations in the enthalpy H The enthalpy is defined by the relation:

where C p is the specific heat at constant pressure

the following relations apply to all fluids

No assumptions have been made concerning the properties of the system and, therefore,

Fluids may be classified in two different ways; either according to their behaviour under the action of externally applied pressure, or according to the effects produced by the action of a shear stress

If the volume of an element of fluid is independent of its pressure and temperature, the fluid is said to be incompressible; if its volume changes it is said to be compressible

No real fluid is completely incompressible though liquids may generally be regarded as such when their flow is considered Gases have a very much higher compressibility than liquids, and appreciable changes in volume may occur if the pressure or temperature is altered However, if the percentage change in the pressure or in the absolute temperature

is small, for practical purposes a gas may also be regarded as incompressible Thus, in practice, volume changes are likely to be important only when the pressure or temperature

of a gas changes by a large proportion The relation between pressure, temperature, and volume of a real gas is generally complex though, except at very high pressures the behaviour of gases approximates to that of the ideal gas for which the volume of a given mass is inversely proportional to the pressure and directly proportional to the absolute temperature At high pressures and when pressure changes are large, however, there may

be appreciable deviations from this law and an approximate equation of state must then

be used

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